Computation of Electric and Magnetic Fields Generated by Cloud-to-Cloud Lightning Channels

Abstract

The paper presents analytical formulas for computation in the time domain of electromagnetic (EM) fields generated by tortuous cloud-to-cloud (CC) lightning channels over a perfectly conducting ground. For the first time, the study was not limited to a horizontal lightning path but was extended to take into account the natural, tortuous geometry of the lightning channel. After the calculation of the step response, a convolution integration was applied for the computation of the fields generated by an arbitrary current source. The produced electric and magnetic fields were then compared with the fields generated by a horizontal channel. The method can be of primary importance to evaluating the hazards for electric and electronic systems of flying aircraft, estimating the voltages induced on overhead transmission lines by CC lightning, and, in general, evaluating the induced effects on sensitive electric and electronic components. Moreover, it may represent a simple, robust, and time-saving tool for estimating important physical parameters that characterize lightning phenomena.

1. Introduction

Lightning is a natural, complex, and dramatic phenomenon that has long had an influence on human psychology and behaviour. The electrical nature of both thunderclouds and lightning was proven by Franklin in 1751 [1]. However, major breakthroughs in lightning research mainly came with the pioneering work of Wilson in regards to the charge structure in clouds [2] and with the invention of the strike camera by Boys in 1926 [3], which produced evidence of the leader channel propagation in cloud-to-ground (CG) lightning [4]. The interest in studies on lightning discharge then grew exponentially: in particular, the discharges occurring between clouds and ground (the so-called cloud-to-ground lightning discharges) were (and are) extensively studied due to their implications on human injuries [5,6], ignition of forest fires, and damage and heavy disturbance to energy and communication transmission lines [7,8,9] and to electric and electronic devices [10,11,12,13], among other reasons. Conversely, although the most frequent lightning phenomena (about 75 percent of the total number of events) occur inside clouds, between clouds, or between clouds and air [14,15] little data are available on cloud discharges due to the inherent difficulty in collecting them and due to the limited hazard on humans, ground objects, and systems. However, in the last two decades, greater attention to cloud discharges has resulted in increased interest due to their potential direct or indirect effects on aircraft and spacecraft [16,17], especially when, for weight reduction and fuel savings, the metallic structure of aeroplanes and avionic systems is replaced by carbon fibre-reinforced composites [18] which provide lower electromagnetic shielding [19,20]. Furthermore, recent studies have provided evidence for the indirect effects of cloud-to-cloud (CC) lightning on overhead medium voltage lines which causes line faults due to inadequate creepage distance of line insulators during wet conditions [21]. Moreover, it should be noted that information concerning the evolution of lightning phenomena, such as the indirect derivation of lightning physical parameters (i.e., the amplitude, shape, and speed of the propagating current or the charge deposition along the channel, etc.) can be obtained from the analysis of EM field waveforms [22]; thus far, the estimation of such parameters has been generally obtained by analysing vertical cloud-to-ground channels [23], and, its is only recently that channel base current parameters have been deduced from EM field waveforms produced by inclined channels [24].

For these reasons, a tool for the computation of the electromagnetic (EM) fields generated by cloud-to-cloud discharges can be of primary importance. It could help to evaluate the hazards of electric and electronic systems on earth and on aircraft or to predict the EM-induced transients in transmission lines. Typical examples of the application of such tools can be found in [25], in which the authors present the Lightning Induced Over-Voltage (LIOV) code, a computer program that uses the EM fields derived by Uman et al. [26] for the calculation of induced overvoltages when a vertical lightning strike hits the ground in proximity to an overhead transmission line. Another example can be found in [27], in which the authors use the Circuit for Lightning Induced Voltage (CiLIV) software for the calculation of voltages induced in complex network configurations by tortuous cloud-to-ground lightning. However, the analytical expression of the fields can also be implemented in specific software in order to support the design of effective lightning protection systems or the sizing of electrical components and electrical devices such as cables, transformers, and PV panels [28,29,30,31,32,33,34]. Last, but not least, the knowledge of the fields could help to acquire new insights into the physical mechanism of lightning by including in the analysis not only the data related to cloud-to-ground phenomena but also those obtained from a much wider class of events that include cloud discharges.

In this context, it is important to also mention the rapid development of machine learning (ML) algorithms able to predict data from lightning phenomena. These algorithms [35], which have been successfully adopted for solving real world problems in several fields, such as medicine, finance, and industry, have also been introduced in the field of lightning for their ability in modelling nonlinear systems and making predictions from a set of training data. For instance, in [36], the authors, by using the results of laboratory measurements, describe an artificial neural network (ANN) able to predict the fields radiated by electrostatic discharges. In [37], ANN is used for the evaluation of the maximum voltage induced by lightning in cable sheaths as a function of different parameters, such as the cable length and the strike location. Recently, a machine learning approach was used to classify lightning discharges and to distinguish between cloud-to-ground and intracloud lightning [38]. The main drawback of such methods, however, is the processing of input data and the resulting computational effort.

The present paper aims to contribute to the topic with the following original contributions: (a) provide analytical expressions of the EM fields produced by a tortuous cloud-to-cloud channel and (b) compare the generated EM fields with those obtained by taking into a account a much simpler horizontal path. In particular, starting from a previously developed model for EM fields calculations, which until now has been used only for cloud-to-ground (CG) lightning [39], the authors extend it in order to take into account the tortuous nature of CC lightning paths, composed of a number of channels with variable slope and height above the ground [14]. In fact, to the best of the authors’ knowledge, such a topic has not yet been studied in full detail: in a pioneering work by Price et al. [40], the computation of fields was limited to a horizontal CC lightning discharge, while in later articles, tortuous [41,42] or inclined channels [43] were considered only in the case of CG discharges. Moreover, it should be taken into account that, with respect to recent approaches [44,45] that are limited to CG lightning, the proposed method has the great advantage of being characterized by a much simpler formulation.

In the first step of this research, as will be clearer in what follows, the model assumes the presence of a perfectly conducting (PEC) ground. This assumption has the great advantage of obtaining simple analytical formulas in the time domain, which require a very short computational time, as will be shown in the following section. Nevertheless, it is well known that the horizontal component of the electric field is affected by the finite conductivity of the terrain. However, it is also known that the calculation requires either a high computational effort [46,47] or the use of approximated formulas [48,49,50]. Moreover, the fact that all proposed approaches are limited to the fields produced by a vertical cloud-to-ground lightning channel and that no formulations are available for cloud-to-cloud discharges should be considered. The results derived in the present study will therefore be useful for a acquiring a deeper understanding of cloud-to-cloud lightning phenomena and of their related effects and will form the basis of future activity that will additionally include the presence of a lossy ground. The major advantages of this kind of approach are summarized as follows: robustness, simplicity of implementation, and time-saving.

The remainder of the paper is organized as follows: Section 2 describes the mathematical model for calculating the fields; Section 3 is dedicated to the validation of the model; in Section 4, the electric and magnetic fields produced by an pulse current flowing along a tortuous lightning path are calculated and compared with the fields produced by a horizontal channel; and Section 5 concludes the article.

2. Electromagnetic Fields Generated by Cloud-to-Cloud Lightning

The lightning channel adopted for the simulation is represented by a number N of arbitrarily oriented segments $C_k$ of negligible cross-section above a perfectly conducting ground. The channel starts with segment $C_1$ from cloud $\#1$ at a generic position and ends up with segment $C_N$ at cloud $\#2$ (Figure 1a); it is traversed by a unit step current that propagates at a constant speed v without attenuation or distortion. Each segment composing the channel can be treated separately: if the EM fields radiated by the generic segment $C_k$ are calculated, then the resulting fields can be computed by summing up the contributions of all the segments. It is worth noting that since the hypothesis of a perfectly conducting ground is adopted, the total fields must also include the contribution of the “image” channel. The electric E(t) and magnetic H(t) fields, radiated by a single segment, can be easily determined if instead of the original Cartesian coordinate system $R^{″}$ (Figure 1a), a proper cylindrical coordinate system $R_c$ is adopted in which the origin is situated at the starting point of the channel and the z-axis is coincident with the direction of the channel (Figure 1b).

In the case of a cylindrical coordinate system, the generic channel segment $C_k$ becomes a single vertical line radiator of height h, and the EM fields at observation point P($r,\phi ,z$) can be easily derived (see Figure 2). For detailed calculation, in order to derive the analytical expressions in the time domain of both the electric and magnetic field, the reader can refer to [39,51]). In order to clarify the solution method, a step-by-step procedure is described in the Appendix A.

In particular, E(t) only has the components $E_r\left(t\right)$ and $E_z\left(t\right)$, while $E_{\phi }\left(t\right)=0$. If $t=0$ is the time when the step current arrives at the generic channel segment; the expressions of the fields are as follows:

$$\left\{\begin{array}{c}{E}_r=\frac{r\left[-tu\left(t-\frac{\sqrt{r^2+z^2}}{c}\right)\right]}{4\pi {\epsilon }_0{\left(\sqrt{r^2+z^2}\right)}^3}+\frac{r\left[\left(t-\frac{h}{v}\right)u\left(t-\frac{h}{v}-\frac{\sqrt{r^2+{(z-h)}^2}}{c}\right)\right]}{4\pi {\epsilon }_0{\left(\sqrt{r^2+{(z-h)}^2}\right)}^3}+I_g\left(t\right)+Eq{1}_r\left(t\right)+Eq{2}_r\left(t\right)\hfill \\ E_{\phi }=0\hfill \\ E_z=\frac{z\left[-tu\left(t-\frac{\sqrt{r^2+z^2}}{c}\right)\right]}{4\pi {\epsilon }_0{\left(\sqrt{r^2+z^2}\right)}^3}+\frac{(z-h)\left[\left(t-\frac{h}{v}\right)u\left(t-\frac{h}{v}-\frac{\sqrt{r^2+{(z-h)}^2}}{c}\right)\right]}{4\pi {\epsilon }_0{\left(\sqrt{r^2+{(z-h)}^2}\right)}^3}+I_p\left(t\right)+Eq{1}_z\left(t\right)+Eq{2}_z\left(t\right)\hfill \end{array}\right.$$

(1)

where $u\left(t\right)$ is the Heaviside step function, c is the speed of light, and v is the velocity of propagation of the step current; all the geometrical parameters are reported in Figure 2. The terms $I_g\left(t\right)$ and $I_p\left(t\right)$ depend on the quantity $z_0\left(t\right)$

$$z_0\left(t\right)=\frac{\frac{t}{v}-\frac{z}{c^2}-\sqrt{{\left(\frac{t}{v}-\frac{z}{c^2}\right)}^2-\left(\frac{1}{v^2}-\frac{1}{c^2}\right)\left(t^2-\frac{r^2}{c^2}-\frac{z^2}{c^2}\right)}}{\left(\frac{1}{v^2}-\frac{1}{c^2}\right)}$$

(2)

and have the following values:

$$I_g\left(t\right)=\left\{\begin{array}{cc}0\hfill & \hfill \mathrm{if}{z}_0\left(t\right)<0\\ \frac{1}{4\pi {\epsilon }_{0}v}\left[\frac{sin\left({\phi }_2\right)-sin\left({\phi }_1\right)}{r}+\frac{r}{c{R}_0^2\left|-\frac{1}{v}-\frac{\left(z_0\left(t\right)-z\right)}{c{R}_0}\right|}\right]\hfill & \hfill \mathrm{if}0\le z_0\left(t\right)\le h\\ \frac{1}{4\pi {\epsilon }_{0}v}\left[\frac{sin\left(arctg\left(\frac{h-z}{r}\right)\right)-sin\left(arctg\left(\frac{-z}{r}\right)\right)}{r}\right]\hfill & \hfill \mathrm{if}{z}_0\left(t\right)>h\end{array}\right.$$

(3)

$$I_p\left(t\right)=\left\{\begin{array}{cc}0\hfill & \hfill \mathrm{if}{z}_0\left(t\right)<0\\ \frac{1}{4\pi {\epsilon }_{0}v}\left[\frac{cos\left({\phi }_2\right)-cos\left({\phi }_1\right)}{r}+\frac{\left(z-z_0\left(t\right)\right)}{c{R}_0^2\left|-\frac{1}{v}-\frac{\left(z_0\left(t\right)-z\right)}{c{R}_0}\right|}\right]\hfill & \hfill \mathrm{if}0\le z_0\left(t\right)\le h\\ \frac{1}{4\pi {\epsilon }_{0}v}\left[\frac{cos\left(arctg\left(\frac{h-z}{r}\right)\right)-cos\left(arctg\left(\frac{-z}{r}\right)\right)}{r}\right]\hfill & \hfill \mathrm{if}{z}_0\left(t\right)>h\end{array}\right.$$

(4)

where

$$\left\{\begin{array}{c}{\phi }_2=arctg\left(\frac{z_0\left(t\right)-z}{r}\right)\hfill \\ {\phi }_1=arctg\left(\frac{-z}{r}\right)\hfill \\ R_0=\sqrt{r^2+{\left(z-z_0\left(t\right)\right)}^2}\hfill \end{array}\right.$$

(5)

The terms $[Eq{1}_r\left(t\right),Eq{1}_z\left(t\right)]$ and $[Eq{2}_r\left(t\right),Eq{2}_z\left(t\right)]$ in Equation (1) represent the contributions to the electric field produced by the charge variation at the bottom and at the upper end of the channel, respectively. Their values are as follows:

$$\left\{\begin{array}{c}Eq{1}_r=\frac{-tu\left(t-\frac{R_1}{c}\right)cos\left({\alpha }_1\right)}{4\pi {\epsilon }_0{R}_1^2}\hfill \\ Eq{1}_z=\frac{-tu\left(t-\frac{R_1}{c}\right)sin\left({\alpha }_1\right)}{4\pi {\epsilon }_0{R}_1^2}\hfill \end{array}\right.$$

(6)

$$\left\{\begin{array}{c}Eq{2}_r=\frac{\left(t-\frac{h}{v}\right)u\left(t-\frac{R_2}{c}-\frac{h}{v}\right)cos\left({\alpha }_2\right)}{4\pi {\epsilon }_0{R}_2^2}\hfill \\ Eq{2}_z=\frac{\left(t-\frac{h}{v}\right)u\left(t-\frac{R_2}{c}-\frac{h}{v}\right)sin\left({\alpha }_2\right)}{4\pi {\epsilon }_0{R}_2^2}\hfill \end{array}\right.$$

(7)

Moreover, as already explicitly remarked, the contributions of the “image” channel and of “image” charges also have to be introduced.

As concerns as the magnetic field H(t), due to the symmetry of the considered geometry, the only component different from zero is $H_{\phi }\left(t\right)$. The expressions of the fields are as follows:

$$\left\{\begin{array}{c}{H}_r=0\hfill \\ H_{\phi }=I_f\left(t\right)\hfill \\ H_z=0\hfill \end{array}\right.$$

(8)

in which, again, the quantity $I_f\left(t\right)$ depends on $z_0\left(t\right)$ in Equation (2):

$$I_f\left(t\right)=\left\{\begin{array}{cc}0\hfill & \hfill \mathrm{if}{z}_0\left(t\right)<0\\ \frac{1}{4\pi }\left[\frac{sin\left({\phi }_2\right)-sin\left({\phi }_1\right)}{r}+\frac{r}{c{R}_0^2\left|-\frac{1}{v}-\frac{\left(z_0\left(t\right)-z\right)}{c{R}_0}\right|}\right]\hfill & \hfill \mathrm{if}0\le z_0\left(t\right)\le h\\ \frac{1}{4\pi }\left[\frac{sin\left(arctg\left(\frac{h-z}{r}\right)\right)-sin\left(arctg\left(\frac{-z}{r}\right)\right)}{r}\right]\hfill & \hfill \mathrm{if}{z}_0\left(t\right)>h\end{array}\right.$$

(9)

Moreover, in this case, the contribution of the “image” channel has to be summed.

3. Model Validation

For the validation of the model described in the previous section and used for the calculation of EM fields generated in CC lightning, the results were compared to those reported by Rubinstein et al. in [52]. For this reason, a vertical cloud-to-ground channel was considered, and the electric and magnetic fields at ground were determined at two distances from the channel base, namely $d_1$ = 5 km and $d_2$ = 200 km. The vertical channel is travelled by a step current of amplitude $I_0=30kA$ at a velocity $v=1/3c$. The current propagates along the channel according to the transmission line (TL) model [53]; that is, without either distortion or attenuation. The vertical channel was adopted as a benchmark since there is abundance of results about radiated EM fields; however, we remark that it can be used for validation purposes because in our model, it is treated as a special case of a tortuous channel, composed on N vertical segments of equal length h and placed at different heights above the ground plane. In Figure 3, in the left column, we show, respectively, the time evolution of the electric and magnetic field at $d_1$ = 5 km calculated with the equations described above. The waveforms can be compared with those shown in Figure 3b in [52]: they markedly highlight the accuracy of the proposed model. As concerns as the amplitude of the electric field, we observe that here and in the following, only the variation of the field produced by the lightning current is calculated: the static field produced by the charge distribution on the clouds prior to the lightning event is not taken into account. This is a common method for presenting data in the literature. Additionally, in Figure 3, on the right column, we have also plotted the time evolution of both the electric and magnetic fields computed at $d_2$ = 200 km. Again, the comparison with the waveshapes shown in Figure 3c in the paper by Rubinstein et al. [52] demonstrates the accuracy of the adopted model.

4. Numerical Results and Discussion

After the verification of the accuracy of the proposed method, we assessed the capabilities of the proposed model to compute EM fields generated by cloud-to-cloud lightning. In the following, we present the results of numerical simulations obtained by considering an horizontal channel originating from point $O_c$ of Cartesian coordinates [0,0,4000] m travelled by the impulsive current $i_p\left(t\right)$ at a velocity $v=2\times {10}^7$ m/s. The analytical expression of the current is as follows:

$$i_p\left(t\right)=2.0·{10}^4·1.935·\left[(e^{(-5·{10}^{4}t)}-e^{(-2·{10}^{5}t)})+\frac{(e^{(-5·{10}^{3}t)}-e^{(-2·{10}^{4}t)})}{3}\right]$$

(10)

The waveshape of the current is the same as the pulse current adopted in [40]; it is shown in Figure 4. It has a peak amplitude of 20 kA. Please note that as reported in the correction in [40], this is the exact expression of the current, while the analytical formula shown in [40] contains a mistake due to a typographical error. The channel is 2 km long and is located in the $yz$ plane. The selected characteristics of the channel, the waveform of the travelling current, and its velocity were thee the same as those used by Price et al. in their pioneering paper [40]. The horizontal channel is composed of $N=20$ horizontal segments of equal length $L_m$ = 100 m. The number of horizontal segments has no effect on the accuracy of the numerical results but influences the computational time since the calculation of the total field is obtained by summing up the fields produced by each segment composing the channel. In this case, the number $N=20$ has been chosen only as a reference value for the evaluation of the required CPU time.

The electric fields were computed at different points above the ground plane at position $P_1\left([0,1000,0]\right)$ and $P_2\left([0,1000,3000]\right)$ in order to take into account the effects of lightning both on objects at ground and on flying objects. Moreover, we also considered the EM fields radiated at the same observation points by a tortuous CC channel in order to highlight the main differences among the fields.

The tortuous path was obtained by randomly defining the number and position of the segments composing it since no 3D data are available in the literature regarding cloud-to-cloud lightning channels. Its average height above the ground is 4000 m, the minimum height is 3950 m, and the maximum height is 4050 m. The tortuous CC channel is composed of $N=74$ segments whose mean length is $L_m$ = 35 m. In order to clarify the considered geometry, the channels and the observation points are shown in Figure 5. In the authors’ opinion, the presence of an “artificial” channel cannot be considered as a limitation because the proposed paper is not focused on presenting a new theoretical model for cloud-to-cloud lightning but rather on proposing a model for EM field calculation.

The fields were computed in a time window of 100 $\mathsf{\mu }$s with a time step $dt$ = 2 ns, resulting in a total number of 50,000 samples for each component of both the electric and magnetic fields. The required computational time for the calculation of the step response, obtained by implementing all calculations in a ®MATLAB environment and by using a Microsoft Window 11 personal computer equipped with 16 GB of RAM and an Intel I7-8700 3.2 GHz processor, was about $t_s$ = 2.6 s for the calculation of the fields produced by the horizontal channel and about $t_s$ = 7.5 s in the case of the tortuous path. The convolution integration was performed in about $t_c$ = 0.3 s for both the horizontal channel and for the tortuous one. As concerns as the observation point $P_1$, due to the hypothesis of a perfectly conductive ground, the horizontal component of the electric field in the reference system $R^{″}$ is always zero. Figure 6 shows the vertical component $E_{z^{″}}\left(t\right)$ created by the horizontal (red) and tortuous (blue) lightning channels. The effect of tortuosity is clearly visible in the jagged shape of the electric field waveform, which presents “humps” following the initial peak. Every kink in the lightning channel has, as a consequence, a variation in the direction of propagation of the current and, accordingly, a rapid change in the electric field: as a result, the fine structure of the electric field waveform reflects the tortuosity of the channel.

Similar conclusions can be drawn from the analysis of the time evolution of the horizontal components $H_{x^{″}}$ and $H_{y^{″}}$ of the magnetic field shown in Figure 7. In particular, the lightning current flowing along a tortuous channel gives rise to a $H_{y^{″}}$ component which has the same order of magnitude of the $H_{x^{″}}$ component. Conversely, due to the geometrical symmetry, the $H_{y^{″}}$ component of the magnetic field produced by an horizontal path is always zero. Furthermore, we can observe that the fine structure in the magnetic field waveforms is more pronounced than in those in the electric field, probably due to the absence of the “masking” effect on the fields produced by the static charge variation on the clouds. The $H_{y^{″}}$ component also presents a zero-crossing and a polarity reversal, which is due to the geometry of the channel. The vertical component of the magnetic field is not shown for both channels because it is always negligible due to the relative position of the observation point (situated in the $yz$ plane) with respect to the lightning path (which mainly evolves in the same plane).

When considering the second observation point $P_2[0,1000,3000]$, we can notice that both channels produce a vertical ($E_{z^{″}}$) and a horizontal ($E_{y^{″}}$) component of the electric field Figure 8. Moreover, in the case of a tortuous path, the $E_{x^{″}}$ component also arises. However, the fine structure of the field is not evident probably because, at a closer distance from the channel, the effect of kinks is masked by the variation of static charge on the cloud. As concerns as the magnetic field, differently from the case of an horizontal channel, all the Cartesian components of the field are present, and their amplitude is in the same order magnitude. This is an important effect to consider because it proves that the EM shielding performance of devices located above the ground can be discernibly affected by the geometrical structure of the channel. In this case, it is also important to highlight that the fine structure of the field waveforms computed at close distance from the channel is less influenced by the channel tortuosity with respect to further observation points. Further investigations could be carried out by taking into account the effects of other important physical parameters, such as the velocity of the travelling current, its waveform, the propagation mechanism along the channel, the position of the observation point, and the geometry of the lightning path. Nevertheless, this sort of survey is beyond the scope of the present paper and will be the objective of upcoming studies. Moreover, it should be noted that considerable effort is needed to the complete the experimental verification of the model and the tuning of its parameters: given a channel current, the model should be able to reproduce the radiated EM fields. In the case of cloud-to-ground lightning, such a validation is usually performed using the triggered lightning technique [54], according to which lightning is initiated by the launching towards the cloud of a small rocket pulling a metal wire from the ground. Although the technique produces “semiartificial” lightning, it has provided interesting results concerning the lightning electromagnetic environment, return-stroke speed, electric fields in the immediate vicinity of the channel, interaction with ground and grounding objects, etc. [55]. Unfortunately, such a technique has inherent difficulties of application in the case of cloud-to-cloud events. Alternatively, the validation of the theoretical model could be performed by using suitable reduced-scale models able to model systems characterised by structural and topological complexity. A procedure for the evaluation of the scale factor for power distribution systems involved in an indirect-lightning event can be found in [56].

5. Conclusions

The analytical formulas for the computation in the time domain of electromagnetic (EM) fields generated by tortuous cloud-to-cloud (CC) lightning channels over a perfectly conducting ground have been presented, together with a detailed, step-by-step procedure of the calculation for each segment composing a lightning channel. The waveforms of the electric and magnetic field components sensibly differ from those produced by a horizontal path and reflect the tortuosity of the channel. In particular, the fine structure of the fields is evidenced in the time evolution of the magnetic field components, while it is mitigated in the electric field waveforms in proximity to the channel due to the masking effect of the field produced by the static charge variation in the clouds. The presence of the three components of the fields support the notion that the proposed tool is essential for the evaluation of the hazards to electric and electronic devices or for the estimation of the induced voltages produced by CC lightning Work is in progress to derive EM fields signatures as a function of the observation point at different distances, elevations, and azimuth angles. Moreover, a further extension of the research activity will be the introduction to the model of a finitely conducting ground.